1. Field of the Invention
The present invention relates to modeling of pipe networks in subsurface reservoirs, and more particularly to modeling the flow in a reservoir formation having multilateral or multi-branch wells.
2. Description of the Related Art
Multi-branch or multilateral wells (also known as Maximum Reservoir Contact or MRC wells) with several pipes formed off a main or mother bore in a formation in a reservoir formation have increasingly been used in oil reservoirs to produce oil, and to prevent water and gas coning. Designing an MRC well requires a reservoir simulator with a pipe flow option to characterize flow from the reservoir through the pipes of the well. Simulators handling pipe flow have faced many problems. These arose from the distinct flow characteristics of two different media: the porous media (reservoir) and the pipe.
Reservoirs are porous rocks where flow is slow, a few centimeters a day, whereas flow inside the pipe in a well is in comparison very fast, i.e. on the order of a meter per second. There is also strong interaction between the reservoir and the well. The reservoir discharges fluid into the well through perforations along the branches which are on the order of hundreds or thousands of feet in length. Once the fluid enters into the pipe, the fluid moves very quickly in comparison with reservoir flow towards a well location known as a hip of the well where the entire production is collected.
Pressure and flow rate distribution inside the well are very sensitive to several variables: the contribution from reservoir, pressure differences inside the well and the production rate. A fraction of a psi pressure drop in the pipe can cause flow of very large volumes of fluid. Therefore, under these conditions it has generally been difficult to develop a stable pressure distribution inside the pipe in the reservoir for modeling purposes.
Because of the inherent difficulties in solving for flow in both media (reservoir and well) together, reservoir simulators have preferred a decoupled approach. A well pressure distribution was assumed, and used to generate models of influx into the wellbore, and then the models solved for the new well bore pressures. The new well pressures were used as boundary conditions for the reservoir simulator to calculate new influx into the wellbore. The process was continued until the influx values from the reservoir, the well pressures and the reservoir variables did not change. This type of processing has been called sequential algorithms. It was well known that sequential algorithms have time step size limitations for time dependent problems. Despite these limitations, sequential (decoupled) methods have often been used in simulators in the petroleum industry because of their perceived convenience.
A further problem has been that fully coupled solutions were expensive in terms of computation time and usage, and often faced convergence problems due to ill conditioned pipe flow matrices in the model.
Since the pipe flow equations produced ill-conditioned matrices, linearized solutions with techniques such as Newton Raphson iterations would not converge unless a good estimate of the actual solution is given. It was not easy to give such an estimate for complex networks with strong influx from the reservoir.